Problem 35 - Precalculus: Functions and Graphs

Question

Refer to Exercise 86 in Section 2.4 The distance \(D\) (in feet) required for a car to safely stop varies directly with its speed \(S\) (in mi/hr). Use the table to determine an approximate value for \(k\) in the variation formula \(D=k S^{23}\) $$\begin{array}{|ccccccc|} \hline \boldsymbol{s} & 20 & 30 & 40 & 50 & 60 & 70 \\ \hline \boldsymbol{D} & 33 & 86 & 167 & 278 & 414 & 593 \\ \hline \end{array}$$ (b) Check your approximation by graphing both the data and \(D\) on the same coordinate axes.

Step-by-Step Solution

Answer

Approximate \(k\) by averaging \(\frac{D}{S^{23}}\) for all given values, and validate by graphing.

Step 1: Understand the Variation Formula

The problem states that the distance \(D\) varies directly with the speed \(S\). The formula given is \(D = k S^{23}\). This means that for each pair of distance \(D\) and speed \(S\), \(k\) can be found as \(k = \frac{D}{S^{23}}\).

Step 2: Calculate the Value of k for Each Data Pair

Using the provided table of values, calculate \(k\) for each pair of \(S\) and \(D\). For example, when \(S = 20\) and \(D = 33\), \[ k = \frac{33}{20^{23}}.\] Perform similar calculations for the other values of \(S\) and \(D\).

Step 3: Approximate k by Averaging

The value of \(k\) can be approximated by averaging all the calculated \(k\) values from Step 2. This will give a single value of \(k\) that best fits all the data points in the table.

Step 4: Graph the Data and the Function

Plot the given data on a graph with \(S\) on the x-axis and \(D\) on the y-axis. Then, use the approximate value of \(k\) to graph the function \(D = k S^{23}\) on the same axes. Compare the plotted data with the graph of the function to check the accuracy of the approximation.

Key Concepts

Variation Constant

When we talk about direct variation, we are referring to a relationship where one variable changes in proportion to another variable. In the context of this exercise, the stopping distance, denoted by \(D\), varies directly with the speed \(S\) of a car. This means that as the speed increases, the stopping distance increases in proportion. The formula to describe this relationship is \(D = k S^{23}\), where \(k\) is known as the variation constant.
The variation constant \(k\) can be understood as a proportionality factor. It determines how much the stopping distance will change for a unit change in speed raised to the 23rd power. To find \(k\), you divide the distance \(D\) by \(S^{23}\) for each data pair in your table. This gives us a unique \(k\) for each pair of data, but the challenge is to find a consistent approximation for \(k\) that fits all data pairs well.

• Calculate \(k\) for each data point using \( k = \frac{D}{S^{23}} \).
• Use these values to find an average \(k\), resulting in the best approximation for the dataset.
• The closer this value fits, the better it models the real-world variation.

Graphing Data

Graphing can be an incredibly useful tool in mathematics, especially when working with variations. Once we calculate our approximate \(k\), we can graph the function \(D = kS^{23}\) to visually compare it with the real-world data we have.
To do this, plot each point from your table where the x-axis represents speed \(S\) and the y-axis represents distance \(D\). Once you've plotted all the points, plot the curve for the function using the approximated \(k\). This helps in visualizing how well the theoretical model matches the observed data.

• Start by plotting each \( (S, D) \) pair on the graph.
• Superimpose the graph of \(D = kS^{23}\) using the approximate \(k\).
• Observe how closely the graph of the equation aligns with the actual data points.

This visual validation is crucial. It allows you to see discrepancies and assess whether adjustments in the function's parameters are necessary for a more accurate model.

Approximating Constants

When approximating constants, it's essential to ensure that the value closely depicts the relationship between variables as described by your data. As detailed in the solutions, first, you calculate \(k = \frac{D}{S^{23}}\) for each individual data point.
But, there is natural variation in real-world data which makes \(k\) numbers different for each data point. To handle this, average all these values of \(k\). Doing so provides a single approximation that can apply to the entire dataset. This average will smooth out noise or discrepancies in the data, providing a balance between overfitting and underfitting your model.

• Calculate individual values of \(k\) for each pair of \( (S, D) \).
• Average these values to find a consistent \(k\) for the dataset.
• This process helps to represent the varied data within a single model as accurately as possible.

Approximating this constant carefully means that when used in your variation equation, \(D = kS^{23}\), it should closely align with real-world data, making it practical and effective for predictions.